Duality and saddle-point type optimality for generalized nonlinear fractional programming

In this paper, we establish two theorems of alternative with generalized subconvexlikeness. We introduce two dual models for a generalized fractional programming problem. Theorems of alternative are then applied to establish duality theorems and a saddle-point type optimality condition.     

Duality in nonlinear programming

In this paper are defined new first- and second-order duals of the nonlinear programming problem with inequality constraints. We introduce a notion of a WD-invex problem. We prove weak, strong, converse, strict converse duality, and other theorems under the hypothesis that the problem is WD-invex. We obtain that a problem with inequality constraints is WD-invex if and only if weak duality holds between the primal and dual problems. We introduce a notion of a second-order WD-invex problem with inequality constraints. The class of WD-invex problems is strictly included in the class of second-order ones. We derive that the first-order duality results are satisfied in the second-order case.     

Duality in disjunctive linear fractional programming

In this note a dual problem is formulated for a given class of disjunctive linear fractional programming problems. This result generalizes to fractional programming the duality theorem of disjunctive linear programming originated by Balas. Two examples are given to illustrate the result.     

Duality in generalized linear fractional programming

We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints. For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established.     

Duality for nonlinear fractional programs

Summary This paper develops duality results for nonlinear fractional programming problems. This is accomplished by using some known results connecting the solutions of a nonlinear fractional program with those of a suitably defined parametric convex program.     

Duality theories in nonlinear semidefinite programming

We consider the duality theories in nonlinear semidefinite programming. Some duality theorems are established to show the important relations among the optimal solutions and optimal values of the primal, the dual and the saddle point problems of nonlinear semidefinite programming.     

Duality in generalized fractional programming via Farkas' lemma

For fractional programs involving several ratios in the objective function, a dual is introduced with the help of Farkas' lemma. Often the dual is again a generalized fractional program. Duality relations are established under weak assumptions. This is done in both the linear case and the nonlinear case. We show that duality can be obtained for these nonconvex programs using only a basic result on linear (convex) inequalities.     

Duality in inexact fractional programming with set-inclusive constraints

This paper extends the fractional programming problem with set-inclusive constraints studied earlier by replacing every coefficient vector in the objective function with a convex set. A dual is formulated, and well-known duality results are established. A numerical example illustrates the dual strategy to obtain the value of the initial problem.The research of the first author was conducted while he was on sabbatical at the Department of Operations Research, Stanford University, Stanford, California. The financial assistance of the International Council for Exchange of Scholars is gratefully acknowledged. The author is grateful to the Department of Operations Research at Stanford for the use of its research facilities.     

Duality on a nondifferentiable minimax fractional programming

We establish the necessary and sufficient optimality conditions on a nondifferentiable minimax fractional programming problem. Subsequently, applying the optimality conditions, we constitute two dual models: Mond-Weir type and Wolfe type. On these duality types, we prove three duality theorems??weak duality theorem, strong duality theorem, and strict converse duality theorem.     

Duality in nondifferentiable multiobjective fractional programming problem with generalized invexity

In this paper, a new class of higher-order (V,α,ρ,θ)-invex function is introduced. Conditions are obtained under which a fractional function is higher-order (V,α,ρ,θ)-invex. Sufficiency of Karush-Kuhn-Tucker conditions is shown under this class of function. We then consider a nondifferentiable multiobjective fractional programming problem and derive the duality theorems.     

Duality in nonlinear multiobjective programming using augmented Lagrangian functions

A vector-valued generalized Lagrangian is constructed for a nonlinear multiobjective programming problem. Using the Lagrangian, a multiobjective dual is considered. Without assuming differentiability, weak and strong duality theorems are established using Pareto efficiency.The research of the second author was partially supported a GTE/SLU grant while visiting St. Lawrence University in the summer of 1991.     

Duality for a class of continuous-time homogeneous fractional programming problems

In this note we present a continuous-time analogue of a duality formulation due to Craven and Mond for a class of homogeneous fractional programming problems. In this duality formulation, the dual problem is also a fractional program with the same objective function as the primal problem.

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Zusammenfassung In dieser Arbeit wird der Dualitätsansatz von Craven und Mond für homogene Quotientenprogramme in endlich vielen Variablen auf unendlich dimensionale Probleme verallgemeinert, wobei Zähler und Nenner sowie die Nebenbedingungsfunktionale als Integrale gegeben sind. Der Ansatz zur Dualität ist dadurch gekennzeichnet, daß das duale Problem wieder ein Quotientenprogramm ist und die gleiche Zielfunktion wie das primale Problem hat.

     

Solving special nonlinear fractional programming problems via parametric linear programming

A new procedure is developed to solve a generalized linear fractional programming problem. We find the optimal solutions in two steps. First we solve a parametric linear programming problem. Using the results of this step we then define a simple optimization problem in the second step. This yields an optimal value which together with the results of the parametric analysis provides the optimal solutions of the considered fractional programming problem.     

Duality in multiobjective nonlinear programming involving semilocally convex and related functions

Necessary optimality conditions are established for a multiobjective nonlinear programming problem involving semilocally convex and related functions in terms of their right differentials. Wolfe type and Mond-Weir type duals are formulated and duality results are proved under the assumptions of semilocal convexity, semilocal quasiconvexity and semilocal pseudoconvexity.     

Complex nonlinear programming: Duality with invexity and equivalent real programs

Necessary conditions for optimality in polyhedral-cone constrained nonlinear complex programs are shown to be sufficient under the assumption of a particular form of invexity. Duality results are thus extended for a Wolfe-type dual. The formulation of real programs equivalent to complex programs via generating matrices is presented.     

Duality theory in multiobjective programming

In this paper, a multiobjective programming problem is considered as that of finding the set of all nondominated solutions with respect to the given domination cone. Two point-to-set maps, the primal map and the dual map, and the vector-valued Lagrangian function are defined, corresponding to the case of a scalar optimization problem. The Lagrange multiplier theorem, the saddle-point theorem, and the duality theorem are derived by using the properties of these maps under adequate convexity assumptions and regularity conditions.